Polynomial Convexity

Preface.

Introduction. Polynomial convexity. Uniform algebras. Plurisubharmonic fuctions. The Cauchy-Fantappiè Integral. The Oka—Weil Theorem. Some examples. Hulls with no analytic structure.-

Some General Properties of Polynomially Convex Sets. Applications of the Cousin problems. Two characterizations of polynomially convex sets. Applications of Morse theory and algebraic topology. Convexity in Stein manifolds.-

Sets of Finite Length. Introduction. One-dimensional varieties. Geometric preliminaries. Function-theoretic preliminaries. Subharmonicity results. Analytic structure in hulls. Finite area. The continuation of varieties.-

Sets of Class A₁. Introductory remarks. Measure-theoretic preliminaries. Sets of class A₁. Finite area. Stokes’s Theorem. The multiplicity function. Counting the branches.-

Further Results. Isoperimetry. Removable singularities. Surfaces in strictly pseudoconvex boundaries.-

Approximation. Totally real manifolds. Holomorphically convex sets. Approximation on totally real manifolds. Some tools from rational approximation. Algebras on surfaces. Tangential approximation.-

Varieties in Strictly Pseudoconvex Domains. Interpolation. Boundary regularity. Uniqueness.-

Examples and Counter Examples. Unions of planes and balls. Pluripolar graphs. Deformations. Sets with symmetry.-

Bibliography. Index.